The Sherrington-Kirkpatrick model for spin glasses: A new approach for the solution

TL;DR

This paper introduces a novel approach using the distributional zeta-function method to analyze the Sherrington-Kirkpatrick spin glass model, providing new analytical insights into its free energy, phase structure, and critical behavior.

Contribution

The paper presents an exact series-based solution for the SK model, revealing the multivalley structure and metastable states, and derives analytical expressions for order parameters near the critical point.

Findings

Agreement of order parameters with phenomenological results

Singular behavior of susceptibility at critical temperature

Ground-state entropy approaches zero at zero temperature

Abstract

We discuss the Sherrington-Kirkpatrick mean-field version of a spin glass within the distributional zeta-function method (DZFM). In the DZFM, since the dominant contribution to the average free energy is written as a series of moments of the partition function of the model, the spin-glass multivalley structure is obtained. Also, an exact expression for the saddle points corresponding to each valley and a global critical temperature showing the existence of many stables or at least metastables equilibrium states is presented. Near the critical point we obtain analytical expressions of the order parameters that are in agreement with phenomenological results. We evaluate the linear and nonlinear susceptibility and we find the expected singular behavior at the spin-glass critical temperature. Furthermore, we obtain a positive definite expression for the entropy and we show that ground-state entropy tends to zero as the temperature goes to zero. We show that our solution is stable for each term in the expansion. Finally, we analyze the behavior of the overlap distribution, where we find a general expression for each moment of the partition function.